He made an unlikely anti-racist campaigner, but there were few voices more critical in the demand for justice for Stephen Lawrence than Paul Dacre and the Daily Mail. It was the Mail's 1997 front page headline, branding Lawrence's alleged killers "Murderers", that helped make the case impossible to ignore. It was, without question, the Mail's finest hour.

The reminder is timely, after a year in which the reputation of tabloid newspapers has been battered. The phone-hacking scandal, and the subsequent Leveson inquiry which resumes next Monday, has painted a picture of a press that has slipped out of the gutter and into the sewer. Anyone who took the Channel 4 satire "Hacks" as a truthful depiction of tabloid life will have come away convinced that this is a boil on society best removed, agreeing with Steve Coogan that any industry that relies on hacking phones and stalking celebs to make a living doesn't deserve to survive a day longer.

Nor have the tabloids done much to win public favour following the hacking revelations. Both Dacre and former Sun editor Kelvin MacKenzie told Leveson he had no business poking his nose into their affairs – MacKenzie in language so coarse he later had to apologise – while tabloid interest in the mother of Hugh Grant's child, with paparazzi camped outside her home, hardly suggests an industry that has got the message and knows it has to change its ways.

And yet in this desire to punish the worst excesses of tabloid journalism, we could lose something valuable. In our determination to throw out badly fouled bathwater, we could dump a much-needed baby.

Those who never pick up a red-top paper, except with a pair of sterilised tongs, might not realise it, but there is more to Britain's tabloids than sleaze and celebrity. Witness the copy of the Daily Mirror I pulled at random from a tottering pile of papers in the Guardian newsroom. It was the edition of 30 November 2011. George Osborne was on the front, with a story on the autumn statement. Most of page two was devoted to a set of 100 Picasso etchings donated to the British Museum, seven of them reproduced. Then three more pages on Osborne, followed by a full-page preview of that day's public sector strike. On the next page, a report on the looting of the British embassy in Tehran, with a short story on, as it happens, the Stephen Lawrence trial. The first celeb items came with the 3am gossip column spread, on pages 12 and 13. Later there was room for a two-page feature on the death of Stalin's daughter.

Maybe that was not a typical day. And, admittedly, the Mirror prides itself on offering a more substantial read than its rivals. That's not new. More than a decade ago, the then editor Piers Morgan reckoned readers living in a post-9/11 world would welcome a more serious paper. He signed up a roster of new contributors, including two former Mirror men John Pilger and Paul Foot, and – full disclosure – me. Few of his obituaries mentioned it, but the last British newspaper to give Christopher Hitchens a regular column was Morgan's Daily Mirror.

Sales rose, apparently welcoming the new seriousness, only falling when the paper's opposition to the Iraq war jarred with readers' traditional support for British troops in action.

And it's not just the Mirror. It won't appeal to many Guardian readers, but Trevor Kavanagh writes serious, informed political commentary in the Sun, while it was the dreaded News of the World which revealed the Pakistani spot-fixing scandal, an important and wholly legitimate story.

Sure, all of this comes wrapped with a healthy salting of gossip and titillation. Tabloid editors don't deny that they are in the business of entertaining as well as informing: broadsheet editors, if they are honest, will admit they do the same, albeit by different means (though sport and sex feature regularly in the Guardian's "most viewed" stories online). But one senior executive told me he also believes it is his job to educate his readers, to explain the world in plain, accessible language. Even if that goal is rarely achieved, it is a noble one, one that any true democrat or egalitarian should support. For a true democracy cannot leave knowledge in the hands of the elite few; it has to be spread widely. So, yes, it has made the most gruesome mistakes and, yes, those will require severe remedy – but Britain needs its popular press, now more than ever.

Twitter: @j_freedland

• Article was amended on 4 January 2012. The original said "former Mirror man John Pilger". The description was meant to apply to both men named "former Mirror men" [John Pilger and Paul Foot].